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Interval oscillation theorems for a second-order linear differential equation. (English) Zbl 1069.34049

Summary: Interval oscillation criteria are given for the forced second-order linear differential equation \[ Ly(t)\equiv (p(t) y')'+ q(t)y= f(t),\quad t\in (0,\infty), \] where \(p\), \(q\), \(f\) are locally integrable functions and \(p(t)> 0\), for \(t> 0\). No restriction is imposed on \(f(t)\) to be the second derivative of an oscillatory function. Our results also allow both \(q\) and \(f\) to change sign in the neighborhood at infinity. In particular, we show that all solutions of \(y''+ c(\sin t)y= t^\beta\cos t\) with \(\beta\geq 0\) are oscillatory, for \(c\geq 1.3448\). This improves an estimate given by A. H. Nasr [Proc. Am. Math. Soc. 126, 123–125 (1998; Zbl 0891.34038)] for the linear equation.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A30 Linear ordinary differential equations and systems

Citations:

Zbl 0891.34038
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References:

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